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Relative concavity of ground state energies as functions of a coupling constant
Physics Letters A 170 (1992) 1—4 North-Holland
PHYSICS LETTERS A
Relative concavity of ground state energies as functions of a coupling constant B. Baumgartner Institulfür Theoretische Physik, Universitdt Wien, Dr. KarlLueger-Ring 1, 1010 Vienna, Austria Received23 March 1992; revised manuscript received 15 July 1992; accepted forpublication 24 August 1992 Communicated by A.P. Fordy
Ground state energies ofHamiltonians in nonrelativistic quantum mechanics are analyzed with respect to their second derivative as functions ofa coupling constant. In one-dimensional systems a comparison of two Hamiltonians yields relative convexity of the ground state energies as implications of a relative convexity of the potentials, by way of the relative log-concavity of the ground state wave functions.
1. General considerations We consider Hamiltonians in £f2 (Pa) or M~ HA =
with —
—
(1)
~ + V+ A.W,
and
V
W
(M),
,~2
potential
functions,
~=
~7... 8 2/8x~.In order to concentrate first on the
mathematical structure we assume units where h2/2m= 1. The physical_coordinates have therefore to be identified with ,%/2m/h2xI. In the first two sections the terms 2, W, and 8/8x 1 may be considered as having the dimension of the square root of an energy. Later on in the applications we will switch to the standard units. A general theorem (3.5, 23 in ref. [1]) tells us that the ground state energy EA is a concave function of 2. Common and Martin [21 proved an inequality for = (42 + 1)
+2 2 and if V(r) is either convex or if A. W=1(1+ concave in r2.I )/r Martin and Stubbe [3] proved an inequality for /d2
3
d
if 2 W=l(1+ 1 )/r2 and if the Laplacian of V(r) is
either positive or negative. Both ref. [2] and ref. [3] used an inequality on (d2/dr2) log q( r) [4], the second logarithmic derivative of the ground state wave function q,. In this paper an inequality on d2E/d22 + ~ will be proved using the generalization of ref. [4]: a relative log-concavity of ground state wave functions implied by relative convexity of the potentials [5,6]. In order to enable the application of this inequality, it is necessar~’to find an expression of d2E/dA2 by the perturbing potential and the ground state wave function We assume that the ground state is not degenerate and that the first order of perturbation theory may be applied [1,7,81. Later on one has to restrict the considerations to one-dimensional systems, but the first part can be done in any dimension. The Hellmann—Feynman theorem gives dE~ (2) The ground state wave functions ç~and 0 XA :~ (3) ~
can be chosen as real-valued. The normalization
I~’AII1implies
(4)
So one can write (with the index A for q.’ and x sup-
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 170, number I
PHYSICS LETTERS A
pressed in order not to overburden the formulas) d2E
(5)
W>
tion of d2/dx2, where it is a seif-adjoint operator, there is an identity
~
where we abbreviate
<~W~ço)= <
19 October 1992
(13) a
.
(6)
to be found by partial integration. (The prime denotes differentiation by x.) So we know that ~x’ —xv” vanishes at the boundaries, and we solve the differ-
(7)
ential equation (8):
Using the identity of perturbation theory
.
(H 0 —E)~=(K W> — W)q,
we find a kind of Wronskian relation
F~(x):r=~(x)X’(x)_X(x)~’(x)
(8)
2.
h
V.(çoVx—xVço)=(W-—)ço
This is used to transform (5) with
=J~(x_y)[W(y)_J~2(y)dy.
)Ici> =
J
(14)
.
(x/~)V~ (~VX—XV~) d~x.
(9)
That one is allowed to perform partial integration without having to worry about the vanishing of the boundary terms follows from referring to methods of functional analysis. We write
We use the Heaviside function 9(x—y) instead of denoting the boundary of integration. Now the potential V is split into two parts: One, where we would explicitly know what the perturbation by A W would effect, and an additional term V~,
v= w2 — w’ + v~,
15
and we consider cases where H
(10)
0_E__A*A,
(16)
W’>O.
(11) with any of the appropriate boundary conditions. Since x is in the domain of H0, it is moreover in the form domain ofA*A, which equals the domain of the closed operator A (see sections VIII.1 and VIII.6 in ref. [9]). Inserting (10) into (7) and then into (5) gives the same result as partial integration of (9):
d2E
So we can use the relative log-concavity of ço implied by relative convexity of V~[61: If 2 W” (d ~dx2 W’ ~) V~(x)~0, (17a) —
2
~
—
-~~)
V~(x)~0
,
(17b)
then 2
4
—(log~)”~W’,
2
(18a)
=—2<~J(~Vx—xV~) /q J~>=—2IlAx~I (12)
2. Comparison of one-dimensional systems We consider systems defined on an interval (a, which may be finite, semifinite or infinite, On finite boundaries Dirichiet boundary conditions are imposed, Because ~ and x are in the domain of definib) ~
2
(18b) [6]u is used for what is here termed
—(logço)”~
It is obvious that any amount of W can be added to V~without changing the relative convexity, so the following inequalities will hold not only for A = 0, but for all values of A.. In integrated form, this relative log-concavity means that W— (W> +~‘/ç9 is (a) decreasing, (b) increasing with x. This enables us to apply the Tche-
Volume 170, number 1
PHYSICS LETTERS A
bychev inequality [10]: 1ff1 andf are both decreasing functions, then
19 October 1992
3. Special cases and applications 3.1. Comparison with the harmonic oscillator
(19a) iff1 is decreasing, f2 increasing, then
Consider
~.
(19b)
(25)
2(p) For V(x)>const+~Ix~,e>’O, and if V(x)inis q’ acting bounded on finite domains, perturbation theory can be applied and the ground state is not degenerate. Relative convexity (1 7a) of the potential meanshere
In our case fj (y) = 6’(x—y) is decreasing, and = W— < W> + ~ with (f
HA=—~+V(x)+A~./~x,
2(b)—co2(a)]_—0. 2>=~[ci’
4~[V(x)_1cx2_,/~]>0,
In case (a) we get F~~,(X)=
)> ~ that is
b
= Je(x_y)~’(y)~(y)dy=_~(x)2,
(22)
a
V”~ic. (26) Introducing physical constants, one gets the fol-
in case (b) the reversed inequality, In any case F~~(x) is negative, since W— < W> is increasing (by (16)), and the Tchebychev inequalityThere (19b)follows can beanapplied withf2=W—.. inequality for the square of F~:
lowing theorem.
F~
is relative convex (or concave) compared to the har-
2~~,(x)4(or ~ in case (b)) 9,(x) Inserted into (12):
.
(23)
(24a)
d2EA/d22<—’2’
(24b)
‘—
This is the main result, which may be stated as Theorem. For one-dimensional non-relativistic quantum systems, described by the Hamiltonian d2 —
+
—
monic oscillator with the potential ~,cx~,in the sense that
d2EA/d22>—’2’ -“
~
Theorem. 2 If d2the potential V in the Hamiltonian h = + V(x) + fIX (27)
w2 w’ + v~+2w —
on the interval (a, b), with Dirichiet boundary conditions if a and/or b is finite, the ground state energies E 2 obey the inequalities of relative convexity (24a),ifV 5isrelativeconvexasstatedin (17a),or ofrelative concavity (24b), if V~is relative concave as stated in (I 7b).
V”~K (or~ic),
(28)
then the susceptibility of the ground state obeys the inequality ~
> -~(or dfI2~ K \ —
~
—
(29)
~
K)
3.2. Comparison with the hydrogen atom The Hamiltonian is defined on P +~ with Dirichiet boundary conditions at x= 0. 2 1(1+1) + V(x) 2= d HA’+ —
~,
.~
V(x) Here
~aconst —
Z/x and bounded on finite domains.
3
Volume 170, number I
PHYSICS LETTERS A
2— W’=l(l+ I )/x2,
W(x) = — (1+1 )/x, W”/W’=—2/x,
W
(31)
so relative convexity means (L~V)(x)=V”(x)+(2/x)V’(x)~O and implies
It should be possible to observe the validity of this inequality in the binding energies ofelectrons at donors or acceptors in semiconductors. There V(x) is the potential generated by the background electrons in the bands, and ~ V> 0.
(32)
References
d2E dZ2
19 October 1992
(33)
2(1+ 1)2
[11W. Thirring, A course in mathematical physics, Vol. 3
(Springer, Berlin, 1979).
Introducing physical constants, h2 /d2 1(l+l)\ H,~=— — (,j~-7— ~ 2m
[2] AK. Common and A. Martin, Europhys. Lett. 4 (1987)
)
1349. [3] A. Martin and J. Stubbe, Europhys. Lett. 14(1991)287. [4] B. Baumgartner, H. Grosse and A. Martin, Phys. Lett. B 146 (1984) 363.
Ze2/4ire +
0
V(x) —
(34)
X
one gets the following theorem. Theorem. If the Laplacian of V(x) at x>0 is greater than zero (or smaller), then 4 me (4~eh)2(/+ 1)2 (or ~) —
4
.
[5] B. Baumgartner, Ann. Phys. 168 (1986) 484. [6] B. Baumgartner, Phys. Lett. A 148 (1990) 443. [7] T. Kato, Perturbation theory for linear operators (Springer, Berlin, 1966). [81 M. Reed and B. Simon, Methods of modem mathematical physics, IV (Academic Press, New York, 1979). [91M. Reed and B. Simon, Methods of modem mathematical [10] G.H. physics, Hardy, I (Academic I.E. Littlewood Press, New and York, G. Polya, 1972). Inequalities (Cambridge Univ. Press, Cambridge, 1959).
PHYSICS LETTERS A
Relative concavity of ground state energies as functions of a coupling constant B. Baumgartner Institulfür Theoretische Physik, Universitdt Wien, Dr. KarlLueger-Ring 1, 1010 Vienna, Austria Received23 March 1992; revised manuscript received 15 July 1992; accepted forpublication 24 August 1992 Communicated by A.P. Fordy
Ground state energies ofHamiltonians in nonrelativistic quantum mechanics are analyzed with respect to their second derivative as functions ofa coupling constant. In one-dimensional systems a comparison of two Hamiltonians yields relative convexity of the ground state energies as implications of a relative convexity of the potentials, by way of the relative log-concavity of the ground state wave functions.
1. General considerations We consider Hamiltonians in £f2 (Pa) or M~ HA =
with —
—
(1)
~ + V+ A.W,
and
V
W
(M),
,~2
potential
functions,
~=
~7... 8 2/8x~.In order to concentrate first on the
mathematical structure we assume units where h2/2m= 1. The physical_coordinates have therefore to be identified with ,%/2m/h2xI. In the first two sections the terms 2, W, and 8/8x 1 may be considered as having the dimension of the square root of an energy. Later on in the applications we will switch to the standard units. A general theorem (3.5, 23 in ref. [1]) tells us that the ground state energy EA is a concave function of 2. Common and Martin [21 proved an inequality for = (42 + 1)
+2 2 and if V(r) is either convex or if A. W=1(1+ concave in r2.I )/r Martin and Stubbe [3] proved an inequality for /d2
3
d
if 2 W=l(1+ 1 )/r2 and if the Laplacian of V(r) is
either positive or negative. Both ref. [2] and ref. [3] used an inequality on (d2/dr2) log q( r) [4], the second logarithmic derivative of the ground state wave function q,. In this paper an inequality on d2E/d22 + ~ will be proved using the generalization of ref. [4]: a relative log-concavity of ground state wave functions implied by relative convexity of the potentials [5,6]. In order to enable the application of this inequality, it is necessar~’to find an expression of d2E/dA2 by the perturbing potential and the ground state wave function We assume that the ground state is not degenerate and that the first order of perturbation theory may be applied [1,7,81. Later on one has to restrict the considerations to one-dimensional systems, but the first part can be done in any dimension. The Hellmann—Feynman theorem gives dE~ (2) The ground state wave functions ç~and 0 XA :~ (3) ~
can be chosen as real-valued. The normalization
I~’AII1implies
(4)
So one can write (with the index A for q.’ and x sup-
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 170, number I
PHYSICS LETTERS A
pressed in order not to overburden the formulas) d2E
(5)
W>
tion of d2/dx2, where it is a seif-adjoint operator, there is an identity
~
where we abbreviate
<~W~ço)= <
19 October 1992
(13) a
.
(6)
to be found by partial integration. (The prime denotes differentiation by x.) So we know that ~x’ —xv” vanishes at the boundaries, and we solve the differ-
(7)
ential equation (8):
Using the identity of perturbation theory
.
(H 0 —E)~=(K W> — W)q,
we find a kind of Wronskian relation
F~(x):r=~(x)X’(x)_X(x)~’(x)
(8)
2.
h
V.(çoVx—xVço)=(W-—
This is used to transform (5) with
=J~(x_y)[W(y)_
J
(14)
.
(x/~)V~ (~VX—XV~) d~x.
(9)
That one is allowed to perform partial integration without having to worry about the vanishing of the boundary terms follows from referring to methods of functional analysis. We write
We use the Heaviside function 9(x—y) instead of denoting the boundary of integration. Now the potential V is split into two parts: One, where we would explicitly know what the perturbation by A W would effect, and an additional term V~,
v= w2 — w’ + v~,
15
and we consider cases where H
(10)
0_E__A*A,
(16)
W’>O.
(11) with any of the appropriate boundary conditions. Since x is in the domain of H0, it is moreover in the form domain ofA*A, which equals the domain of the closed operator A (see sections VIII.1 and VIII.6 in ref. [9]). Inserting (10) into (7) and then into (5) gives the same result as partial integration of (9):
d2E
So we can use the relative log-concavity of ço implied by relative convexity of V~[61: If 2 W” (d ~dx2 W’ ~) V~(x)~0, (17a) —
2
~
—
-~~)
V~(x)~0
,
(17b)
then 2
4
—(log~)”~W’,
2
(18a)
=—2<~J(~Vx—xV~) /q J~>=—2IlAx~I (12)
2. Comparison of one-dimensional systems We consider systems defined on an interval (a, which may be finite, semifinite or infinite, On finite boundaries Dirichiet boundary conditions are imposed, Because ~ and x are in the domain of definib) ~
2
(18b) [6]u is used for what is here termed
—(logço)”~
It is obvious that any amount of W can be added to V~without changing the relative convexity, so the following inequalities will hold not only for A = 0, but for all values of A.. In integrated form, this relative log-concavity means that W— (W> +~‘/ç9 is (a) decreasing, (b) increasing with x. This enables us to apply the Tche-
Volume 170, number 1
PHYSICS LETTERS A
bychev inequality [10]: 1ff1 andf are both decreasing functions, then
19 October 1992
3. Special cases and applications 3.1. Comparison with the harmonic oscillator
(19a) iff1 is decreasing, f2 increasing, then
Consider
(19b)
(25)
2(p) For V(x)>const+~Ix~,e>’O, and if V(x)inis q’ acting bounded on finite domains, perturbation theory can be applied and the ground state is not degenerate. Relative convexity (1 7a) of the potential meanshere
In our case fj (y) = 6’(x—y) is decreasing, and = W— < W> + ~ with (f
HA=—~+V(x)+A~./~x,
2(b)—co2(a)]_—0. 2>=~[ci’
4~[V(x)_1cx2_,/~]>0,
In case (a) we get F~~,(X)=
)> ~ that is
b
= Je(x_y)~’(y)~(y)dy=_~(x)2,
(22)
a
V”~ic. (26) Introducing physical constants, one gets the fol-
in case (b) the reversed inequality, In any case F~~(x) is negative, since W— < W> is increasing (by (16)), and the Tchebychev inequalityThere (19b)follows can beanapplied withf2=W—.
lowing theorem.
F~
is relative convex (or concave) compared to the har-
2~~,(x)4(or ~ in case (b)) 9,(x) Inserted into (12):
.
(23)
(24a)
d2EA/d22<—’2’
(24b)
‘—
This is the main result, which may be stated as Theorem. For one-dimensional non-relativistic quantum systems, described by the Hamiltonian d2 —
+
—
monic oscillator with the potential ~,cx~,in the sense that
d2EA/d22>—’2’ -“
~
Theorem. 2 If d2the potential V in the Hamiltonian h = + V(x) + fIX (27)
w2 w’ + v~+2w —
on the interval (a, b), with Dirichiet boundary conditions if a and/or b is finite, the ground state energies E 2 obey the inequalities of relative convexity (24a),ifV 5isrelativeconvexasstatedin (17a),or ofrelative concavity (24b), if V~is relative concave as stated in (I 7b).
V”~K (or~ic),
(28)
then the susceptibility of the ground state obeys the inequality ~
> -~(or dfI2~ K \ —
~
—
(29)
~
K)
3.2. Comparison with the hydrogen atom The Hamiltonian is defined on P +~ with Dirichiet boundary conditions at x= 0. 2 1(1+1) + V(x) 2= d HA’+ —
~,
.~
V(x) Here
~aconst —
Z/x and bounded on finite domains.
3
Volume 170, number I
PHYSICS LETTERS A
2— W’=l(l+ I )/x2,
W(x) = — (1+1 )/x, W”/W’=—2/x,
W
(31)
so relative convexity means (L~V)(x)=V”(x)+(2/x)V’(x)~O and implies
It should be possible to observe the validity of this inequality in the binding energies ofelectrons at donors or acceptors in semiconductors. There V(x) is the potential generated by the background electrons in the bands, and ~ V> 0.
(32)
References
d2E dZ2
19 October 1992
(33)
2(1+ 1)2
[11W. Thirring, A course in mathematical physics, Vol. 3
(Springer, Berlin, 1979).
Introducing physical constants, h2 /d2 1(l+l)\ H,~=— — (,j~-7— ~ 2m
[2] AK. Common and A. Martin, Europhys. Lett. 4 (1987)
)
1349. [3] A. Martin and J. Stubbe, Europhys. Lett. 14(1991)287. [4] B. Baumgartner, H. Grosse and A. Martin, Phys. Lett. B 146 (1984) 363.
Ze2/4ire +
0
V(x) —
(34)
X
one gets the following theorem. Theorem. If the Laplacian of V(x) at x>0 is greater than zero (or smaller), then 4 me (4~eh)2(/+ 1)2 (or ~) —
4
.
[5] B. Baumgartner, Ann. Phys. 168 (1986) 484. [6] B. Baumgartner, Phys. Lett. A 148 (1990) 443. [7] T. Kato, Perturbation theory for linear operators (Springer, Berlin, 1966). [81 M. Reed and B. Simon, Methods of modem mathematical physics, IV (Academic Press, New York, 1979). [91M. Reed and B. Simon, Methods of modem mathematical [10] G.H. physics, Hardy, I (Academic I.E. Littlewood Press, New and York, G. Polya, 1972). Inequalities (Cambridge Univ. Press, Cambridge, 1959).